Nonlinear Dynamics

, Volume 71, Issue 1–2, pp 175–185 | Cite as

Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities

  • S. B. Li
  • W. Zhang
  • L. J. Gao
Original Paper


Based on temporal rescaling and harmonic balance, an extended asymptotic perturbation method for parametrically excited two-degree-of-freedom systems with square and cubic nonlinearities is proposed to study the nonlinear dynamics under 1:2 internal resonance. This asymptotic perturbation method is employed to transform the two-degree-of-freedom nonlinear systems into a four-dimensional nonlinear averaged equation governing the amplitudes and phases of the approximation solutions. Linear stable analysis at equilibrium solutions of the averaged equation is done to show bifurcations of periodic motion and homoclinic motions. Furthermore, analytical expressions of homoclinic orbits and heteroclinic cycles for the averaged equation without dampings are obtained. Considering the action of the damping, the bifurcations of limit cycles are also investigated. A concrete example is further provided to discuss the correctness and accuracy of the extended asymptotic perturbation method in the case of small-amplitude motion for the two-degree-of-freedom nonlinear system.


Extended asymptotic perturbation method Two-degree-of-freedom systems Square and cubic nonlinearities Periodic motions Homoclinic motions 



The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11072008, 10732020, and No. 11102226, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB), the NSFC Tianyuan Youth Foundation of Mathematics of China through Grant No. 11126136, the Scientific Research Foundation of Civil Aviation University of China through Grant No. 2010QD04X, the Fundamental Research Funds for the Central Universities through Grant No. ZXH2011D006 and No. ZXH2012K004.


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.College of ScienceCivil Aviation University of ChinaTianjinP.R. China
  2. 2.College of Mechanical EngineeringBeijing University of TechnologyBeijingP.R. China
  3. 3.College of Electronic Information EngineeringCivil Aviation University of ChinaTianjinP.R. China

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